Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 11 $
Sign $-0.870 + 0.492i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (1 − 1.41i)3-s i·5-s + (1.41 − 2.00i)6-s i·7-s − 2.82·8-s + (−1.00 − 2.82i)9-s − 1.41i·10-s + (−3 − 1.41i)11-s + 2.24i·13-s − 1.41i·14-s + (−1.41 − i)15-s − 4.00·16-s − 5.82·17-s + (−1.41 − 4.00i)18-s − 3.24i·19-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.577 − 0.816i)3-s − 0.447i·5-s + (0.577 − 0.816i)6-s − 0.377i·7-s − 0.999·8-s + (−0.333 − 0.942i)9-s − 0.447i·10-s + (−0.904 − 0.426i)11-s + 0.621i·13-s − 0.377i·14-s + (−0.365 − 0.258i)15-s − 1.00·16-s − 1.41·17-s + (−0.333 − 0.942i)18-s − 0.743i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-0.870 + 0.492i$
motivic weight  =  \(1\)
character  :  $\chi_{1155} (1121, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1155,\ (\ :1/2),\ -0.870 + 0.492i)\)
\(L(1)\)  \(\approx\)  \(1.892414867\)
\(L(\frac12)\)  \(\approx\)  \(1.892414867\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (3 + 1.41i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 + 5.82T + 17T^{2} \)
19 \( 1 + 3.24iT - 19T^{2} \)
23 \( 1 + 1.24iT - 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 - 8.24T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 + 3.24iT - 43T^{2} \)
47 \( 1 + 1.07iT - 47T^{2} \)
53 \( 1 + 4.41iT - 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 - 9.24iT - 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + 6.48iT - 73T^{2} \)
79 \( 1 + 6.24iT - 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.139758022301646199359626877677, −8.650414389463137057219876824771, −7.83861477618943217929226722440, −6.68341426209116621916871477674, −6.19853898811194708671325459513, −4.91179356007377517280699769106, −4.33492508349302071643485766775, −3.13364371502824762233945121972, −2.28351808249928811457742565149, −0.51968838239840333391418286012, 2.49604253961629437896164921632, 3.00108665063701512194098575358, 4.19384811111398399880030679746, 4.77517801951387177258266818107, 5.66180039340876746857808495419, 6.52321052831688718640739036650, 7.86203151545594919909184161528, 8.475996808158554914037102510333, 9.411180763752445079242197060346, 10.13192991536889851883323599606

Graph of the $Z$-function along the critical line